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The Nodding Sphere and the Bird's Beak: D'Alembert's Dispute with Euler - Logarithms of Negative Numbers

Robert E. Bradley

Euler's paper "On the controversy between Messrs. Leibniz and Bernoulli concerning logarithms of negative and imaginary numbers" [9] is the first publication in which the riddle of logarithms of negative numbers is solved. An English translation of this article by Stacy Langton is available by clicking here. Although d'Alembert, like Johann (I) Bernoulli before him, believed that \(\ln{\left(-x\right)} = \ln{x}\) for positive real numbers \(x,\) Euler showed that \(\ln{\left(-x\right)} = \ln{x}+i(2n+1)\pi\) for any integer \(n.\) Euler and d'Alembert had debated this at great length in their early correspondence, and although Euler was eventually able to persuade his correspondent that the logarithm is a complex-valued multi-function, d'Alembert never abandoned his assertion that \(\ln{x}\) was a value of \(\ln{\left(-x\right)}.\)

In response to Euler's article, d'Alembert addressed a memoir to the Berlin Academy in June of 1752, in which he attacked Euler's position and attempted to prove that ln(-x)=lnx, or at least persuade the reader that it was possible. The Academy never published the paper, presumably on account of both its mathematical content and polemical tone. Instead, d'Alembert published the piece in the first volume of his Opuscules [10], his self-published collection of mathematical papers, which appeared in 8 volumes between 1761 and 1780.

Since d'Alembert disagreed with Euler's conclusions about logarithms in [9], it's not quite fair for Hankins to characterize d'Alembert's concerns about this paper as involving a matter of his priority in a discovery. On the other hand, priority was precisely d'Alembert's concern with regards to Euler's three other pieces in the 1749 volume. To that end, d'Alembert addressed a second essay to the Berlin Academy in June of 1752. Called "Observations on several memoirs printed in the Academy's volume for 1749," [6, p. 337-346] this essay made the case that various results in Euler's papers [11, 12, 13]  were first proved by d'Alembert, and revealed to Euler through letters and pieces already in print.

Robert E. Bradley, "The Nodding Sphere and the Bird's Beak: D'Alembert's Dispute with Euler - Logarithms of Negative Numbers," Loci (July 2012)